On delocalization in the six-vertex model

We show that the six-vertex model with parameter $c\in [\sqrt{3},2]$ on a square lattice torus has an ergodic infinite-volume limit as the size of the torus grows to infinity. Moreover we prove that for $c\in[\sqrt{2+\sqrt 2}, 2]$,theassociatedheightfunctionon$\mathbb Z^2$ has unbounded variance. The proof relies on an extension of the Baxter-Kelland-Wu representation of the six-vertex model to multi-point correlation functions of the associated spin model. Other crucial ingredients are the uniqueness and percolation properties of the critical random cluster measure for $q\in [1,4]$, and recent results relating the decay of correlations in the spin model with the delocalization of the height function.